Hyperbola
Pair of tangents
Grade 11
Question:
<p>The point of intersection of two tangents to the hyperbola \(\frac{x^2}{25} - \frac{y^2}{144} = 1\), the product of whose slopes is \(c^2\), lies on the curve</p>
<p>(a) \(\frac{x^2}{25} - \frac{y^2}{144} = 1\)</p>
<p>(b) \(\frac{x^2}{144} - \frac{y^2}{25} = 1\)</p>
<p>(c) A directrix</p>
<p>(d) Cannot be determined</p>
Step-by-Step Solution
Key Concept: Use the property of tangent slopes and their products to identify the locus of intersection points.
<p><strong>Solution:</strong> For the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) with \(a^2 = 25\), \(b^2 = 144\).</p><p>We have \(c^2 = a^2 + b^2 = 25 + 144 = 169\).</p><p>If two tangents from a point \((h, k)\) have slopes \(m_1\) and \(m_2\) with \(m_1 \cdot m_2 = c^2 = 169\), then the locus of such points is given by the auxiliary circle or director circle.</p><p>For a hyperbola, if the product of slopes of two tangents is \(\frac{b^2}{a^2}\), the intersection lies on the conjugate hyperbola. Since the product equals \(c^2\), the point lies on \(\frac{x^2}{25} - \frac{y^2}{144} = 1\) (the original hyperbola).</p><p>∴ Answer is (a).</p>
Correct Answer: A